On stability and hyperbolicity for polynomial automorphisms of C^2

Abstract : Let (fλ)λ∈Λ be a holomorphic family of polynomial automorphisms of C2. Fol- lowing previous work of Dujardin and Lyubich, we say that such a family is weakly stable if saddle periodic orbits do not bifurcate. It is an open question whether this property is equivalent to structural stability on the Julia set J∗ (that is, the closure of the set of saddle periodic points). In this paper we introduce a notion of regular point for a polynomial automorphism, inspired by Pesin theory, and prove that in a weakly stable family, the set of regular points moves holomorphically. It follows that a weakly stable family is probabilistically structurally stable, in a very strong sense. Another consequence of these techniques is that weak stability preserves uniform hyperbolicity on J∗.
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Pierre Berger, Romain Dujardin. On stability and hyperbolicity for polynomial automorphisms of C^2. Annales scientifiques de l'Ecole normale supérieure, 2017, 50 (2), pp.449-477. ⟨hal-01068578⟩

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